Fractal Geometry and Dynamical Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Geometry".

Deadline for manuscript submissions: closed (31 December 2024) | Viewed by 4078

Special Issue Editors


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Guest Editor
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Brownsville, TX, USA
Interests: quantization; theory and applications; fractal geometry; dynamical systems

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Guest Editor
Indian Institute of Information Technology, Allahabad, India
Interests: fractal geometry; quantization; theory and applications; approximation theory; fixed point theory; applications

Special Issue Information

Dear Colleagues,

Fractal geometry has recently expanded into a vast field that encompasses practically all fields of science and engineering. It is the study of the properties of fractal objects, which are commonly referred to as fractals. The notion of fractal geometry naturally arises in the study of non-linear functions. We know that irregular sets provide a much better representation of natural phenomena than classical geometry. Fractal geometry is a tool to understand the geometric properties of such irregular sets. The area of fractal geometry and dynamical systems is multidisciplinary in nature with a huge scope of future research and real-life applications in signal processing, image processing, computer-aided geometric design, bioscience, physics, etc.

For this Special Issue, we intend to collect research presenting recent advances in the field of fractal geometry and dynamical systems.

Prof. Dr. Mrinal Kanti Roychowdhury
Dr. Saurabh Verma
Guest Editors

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Keywords

  • iterated function systems
  • julia and mandelbrot sets
  • self-similar sets and measures
  • self-affine and self-conformal sets/measures
  • fractal interpolation functions
  • quantization dimension
  • box dimension, hausdorff dimension and Lq dimensions

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Published Papers (1 paper)

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Research

10 pages, 1157 KiB  
Article
Some Insights into the Sierpiński Triangle Paradox
by Miguel-Ángel Martínez-Cruz, Julián Patiño-Ortiz, Miguel Patiño-Ortiz and Alexander S. Balankin
Fractal Fract. 2024, 8(11), 655; https://doi.org/10.3390/fractalfract8110655 - 11 Nov 2024
Viewed by 3611
Abstract
We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that [...] Read more.
We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that SACSG. Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve PC0,12 differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension D=ln3/ln2, their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension dlSAC=d=1, whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, dlSG=D. Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted. Full article
(This article belongs to the Special Issue Fractal Geometry and Dynamical Systems)
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